A path from \(i\)  to \(j\)  is a sequence of distinct vertices starting at \(i\)  and ending at \(j\)  such that consecutive vertices are adjacent. If there is a path between all pairs of vertices of a graph \(\mathcal {G}\) , then \(\mathcal {G}\)  is connected; else \(\mathcal {G}\)  is disconnected. An
induced subgraph of \(\mathcal {G}\)  that is maximal, subject to being connected, is called a connected component of \(\mathcal {G}\) . A cycle is a sequence of adjacent edges without repetition that starts and ends at the same node. A graph with no cycles is called acyclic. A connected and acyclic graph is a tree. A spanning tree subgraph of \(\mathcal {G}\)  is a tree that covers all vertices in \(\mathcal {G}\) . An acyclic graph with multiple connected components is a forest. A spanning forest subgraph of \(\mathcal {G}\)  is a forest that covers all vertices in \(\mathcal {G}\) . Spanning forests may include connected components with a single node. A review of graph theory fundamentals is available in [1]}.
